Optimal. Leaf size=98 \[ \frac{3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac{x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]
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Rubi [A] time = 0.0709716, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac{x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x^n)^(2/n))^(-3),x]
[Out]
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Rubi in Sympy [A] time = 8.313, size = 82, normalized size = 0.84 \[ \frac{x}{4 a \left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )^{2}} + \frac{3 x}{8 a^{2} \left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )} + \frac{3 x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{b} \left (c x^{n}\right )^{\frac{1}{n}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*(c*x**n)**(2/n))**3,x)
[Out]
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Mathematica [A] time = 4.40314, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + b*(c*x^n)^(2/n))^(-3),x]
[Out]
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Maple [F] time = 0.044, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*(c*x^n)^(2/n))^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(2/n)*b + a)^(-3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265094, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a b c^{\frac{2}{n}} x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b c^{\frac{2}{n}} x +{\left (b c^{\frac{2}{n}} x^{2} - a\right )} \sqrt{-a b c^{\frac{2}{n}}}}{b c^{\frac{2}{n}} x^{2} + a}\right ) + 2 \,{\left (3 \, b c^{\frac{2}{n}} x^{3} + 5 \, a x\right )} \sqrt{-a b c^{\frac{2}{n}}}}{16 \,{\left (a^{2} b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a^{3} b c^{\frac{2}{n}} x^{2} + a^{4}\right )} \sqrt{-a b c^{\frac{2}{n}}}}, \frac{3 \,{\left (b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a b c^{\frac{2}{n}} x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right ) +{\left (3 \, b c^{\frac{2}{n}} x^{3} + 5 \, a x\right )} \sqrt{a b c^{\frac{2}{n}}}}{8 \,{\left (a^{2} b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a^{3} b c^{\frac{2}{n}} x^{2} + a^{4}\right )} \sqrt{a b c^{\frac{2}{n}}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(2/n)*b + a)^(-3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*(c*x**n)**(2/n))**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\frac{2}{n}} b + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(2/n)*b + a)^(-3),x, algorithm="giac")
[Out]